Sliced optimal transport: is it a suitable replacement?
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917676786384896 |
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| author | Kitagawa, Jun Takatsu, Asuka |
| author_facet | Kitagawa, Jun Takatsu, Asuka |
| contents | We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced Wasserstein and max-sliced Wasserstein metrics. We then show that these are complete, separable metric spaces that are topologically equivalent to the classical Monge--Kantorovich metrics and these metrics have a dual representation. However, we also prove these sliced metrics are $\textit{not}$ bi-Lipschitz equivalent to the classical ones in most cases, and also the spaces are (except for an endpoint case) $\textit{not}$ geodesic. The completeness, duality, and non-geodesicness are new even in the sliced and max-sliced Wasserstein cases, and non bi-Lipschitz equivalence is only known for a few specific cases. In particular this indicates that sliced and max-sliced Wasserstein metrics are not suitable direct replacements for the classical Monge--Kantorovich metrics in problems where the specific metric or geodesic structure are critical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15874 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sliced optimal transport: is it a suitable replacement? Kitagawa, Jun Takatsu, Asuka Metric Geometry Optimization and Control Probability 49Q22, 44A12 We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced Wasserstein and max-sliced Wasserstein metrics. We then show that these are complete, separable metric spaces that are topologically equivalent to the classical Monge--Kantorovich metrics and these metrics have a dual representation. However, we also prove these sliced metrics are $\textit{not}$ bi-Lipschitz equivalent to the classical ones in most cases, and also the spaces are (except for an endpoint case) $\textit{not}$ geodesic. The completeness, duality, and non-geodesicness are new even in the sliced and max-sliced Wasserstein cases, and non bi-Lipschitz equivalence is only known for a few specific cases. In particular this indicates that sliced and max-sliced Wasserstein metrics are not suitable direct replacements for the classical Monge--Kantorovich metrics in problems where the specific metric or geodesic structure are critical. |
| title | Sliced optimal transport: is it a suitable replacement? |
| topic | Metric Geometry Optimization and Control Probability 49Q22, 44A12 |
| url | https://arxiv.org/abs/2311.15874 |